2023/09/29 by Thanasis Bouganis, Bouganis, Thanasis, Rafail Psyroukis +1
Mathematics · #11F50 #11F55 #11F66 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2309.17237
openalex publication_date 2023/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K = ℚ(i). We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group U5(K), diagonally-restricted on U2(K)× U2(K)× U1(K), against two Hermitian cuspidal eigenforms F, G of degree 2 and an elliptic cuspidal eigenform h (seen as a Hermitian modular form of degree 1), all having weight k ≡ 0 \pmod 4. We obtain, through this consideration, an integral representation of a certain Dirichlet series, together with an additional residue term. By taking F to belong in the Maass space, we are able to show that the Dirichlet series possesses an Euler product. Moreover, its p-factor for an inert prime p can be essentially identified with the twist by h of a degree six Euler factor attached to G by Gritsenko. The question of whether the same holds for the primes that split remains unanswered here, even though we make considerable steps in that direction too. Our paper is inspired by a work of Heim, who considered a similar question in the case of Siegel modular forms.